<?php

    /**
     * @package JAMA
     *    For an m-by-n matrix A with m >= n, the singular value decomposition is
     *    an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
     *    an n-by-n orthogonal matrix V so that A = U*S*V'.
     *    The singular values, sigma[$k] = S[$k][$k], are ordered so that
     *    sigma[0] >= sigma[1] >= ... >= sigma[n-1].
     *    The singular value decompostion always exists, so the constructor will
     *    never fail.  The matrix condition number and the effective numerical
     *    rank can be computed from this decomposition.
     * @author  Paul Meagher
     * @license PHP v3.0
     * @version 1.1
     */
    class SingularValueDecomposition {
        /**
         *    Internal storage of U.
         * @var array
         */
        private $U = [];
        /**
         *    Internal storage of V.
         * @var array
         */
        private $V = [];
        /**
         *    Internal storage of singular values.
         * @var array
         */
        private $s = [];
        /**
         *    Row dimension.
         * @var int
         */
        private $m;
        /**
         *    Column dimension.
         * @var int
         */
        private $n;

        /**
         *    Construct the singular value decomposition
         *    Derived from LINPACK code.
         * @param $A Rectangular matrix
         * @return Structure to access U, S and V.
         */
        public function __construct($Arg) {
            // Initialize.
            $A       = $Arg->getArrayCopy();
            $this->m = $Arg->getRowDimension();
            $this->n = $Arg->getColumnDimension();
            $nu      = min($this->m, $this->n);
            $e       = [];
            $work    = [];
            $wantu   = true;
            $wantv   = true;
            $nct     = min($this->m - 1, $this->n);
            $nrt     = max(0, min($this->n - 2, $this->m));
            // Reduce A to bidiagonal form, storing the diagonal elements
            // in s and the super-diagonal elements in e.
            for ($k = 0; $k < max($nct, $nrt); ++$k) {
                if ($k < $nct) {
                    // Compute the transformation for the k-th column and
                    // place the k-th diagonal in s[$k].
                    // Compute 2-norm of k-th column without under/overflow.
                    $this->s[$k] = 0;
                    for ($i = $k; $i < $this->m; ++$i) {
                        $this->s[$k] = hypo($this->s[$k], $A[$i][$k]);
                    }
                    if ($this->s[$k] != 0.0) {
                        if ($A[$k][$k] < 0.0) {
                            $this->s[$k] = -$this->s[$k];
                        }
                        for ($i = $k; $i < $this->m; ++$i) {
                            $A[$i][$k] /= $this->s[$k];
                        }
                        $A[$k][$k] += 1.0;
                    }
                    $this->s[$k] = -$this->s[$k];
                }
                for ($j = $k + 1; $j < $this->n; ++$j) {
                    if (($k < $nct) & ($this->s[$k] != 0.0)) {
                        // Apply the transformation.
                        $t = 0;
                        for ($i = $k; $i < $this->m; ++$i) {
                            $t += $A[$i][$k] * $A[$i][$j];
                        }
                        $t = -$t / $A[$k][$k];
                        for ($i = $k; $i < $this->m; ++$i) {
                            $A[$i][$j] += $t * $A[$i][$k];
                        }
                        // Place the k-th row of A into e for the
                        // subsequent calculation of the row transformation.
                        $e[$j] = $A[$k][$j];
                    }
                }
                if ($wantu and ($k < $nct)) {
                    // Place the transformation in U for subsequent back
                    // multiplication.
                    for ($i = $k; $i < $this->m; ++$i) {
                        $this->U[$i][$k] = $A[$i][$k];
                    }
                }
                if ($k < $nrt) {
                    // Compute the k-th row transformation and place the
                    // k-th super-diagonal in e[$k].
                    // Compute 2-norm without under/overflow.
                    $e[$k] = 0;
                    for ($i = $k + 1; $i < $this->n; ++$i) {
                        $e[$k] = hypo($e[$k], $e[$i]);
                    }
                    if ($e[$k] != 0.0) {
                        if ($e[$k + 1] < 0.0) {
                            $e[$k] = -$e[$k];
                        }
                        for ($i = $k + 1; $i < $this->n; ++$i) {
                            $e[$i] /= $e[$k];
                        }
                        $e[$k + 1] += 1.0;
                    }
                    $e[$k] = -$e[$k];
                    if (($k + 1 < $this->m) and ($e[$k] != 0.0)) {
                        // Apply the transformation.
                        for ($i = $k + 1; $i < $this->m; ++$i) {
                            $work[$i] = 0.0;
                        }
                        for ($j = $k + 1; $j < $this->n; ++$j) {
                            for ($i = $k + 1; $i < $this->m; ++$i) {
                                $work[$i] += $e[$j] * $A[$i][$j];
                            }
                        }
                        for ($j = $k + 1; $j < $this->n; ++$j) {
                            $t = -$e[$j] / $e[$k + 1];
                            for ($i = $k + 1; $i < $this->m; ++$i) {
                                $A[$i][$j] += $t * $work[$i];
                            }
                        }
                    }
                    if ($wantv) {
                        // Place the transformation in V for subsequent
                        // back multiplication.
                        for ($i = $k + 1; $i < $this->n; ++$i) {
                            $this->V[$i][$k] = $e[$i];
                        }
                    }
                }
            }
            // Set up the final bidiagonal matrix or order p.
            $p = min($this->n, $this->m + 1);
            if ($nct < $this->n) {
                $this->s[$nct] = $A[$nct][$nct];
            }
            if ($this->m < $p) {
                $this->s[$p - 1] = 0.0;
            }
            if ($nrt + 1 < $p) {
                $e[$nrt] = $A[$nrt][$p - 1];
            }
            $e[$p - 1] = 0.0;
            // If required, generate U.
            if ($wantu) {
                for ($j = $nct; $j < $nu; ++$j) {
                    for ($i = 0; $i < $this->m; ++$i) {
                        $this->U[$i][$j] = 0.0;
                    }
                    $this->U[$j][$j] = 1.0;
                }
                for ($k = $nct - 1; $k >= 0; --$k) {
                    if ($this->s[$k] != 0.0) {
                        for ($j = $k + 1; $j < $nu; ++$j) {
                            $t = 0;
                            for ($i = $k; $i < $this->m; ++$i) {
                                $t += $this->U[$i][$k] * $this->U[$i][$j];
                            }
                            $t = -$t / $this->U[$k][$k];
                            for ($i = $k; $i < $this->m; ++$i) {
                                $this->U[$i][$j] += $t * $this->U[$i][$k];
                            }
                        }
                        for ($i = $k; $i < $this->m; ++$i) {
                            $this->U[$i][$k] = -$this->U[$i][$k];
                        }
                        $this->U[$k][$k] = 1.0 + $this->U[$k][$k];
                        for ($i = 0; $i < $k - 1; ++$i) {
                            $this->U[$i][$k] = 0.0;
                        }
                    } else {
                        for ($i = 0; $i < $this->m; ++$i) {
                            $this->U[$i][$k] = 0.0;
                        }
                        $this->U[$k][$k] = 1.0;
                    }
                }
            }
            // If required, generate V.
            if ($wantv) {
                for ($k = $this->n - 1; $k >= 0; --$k) {
                    if (($k < $nrt) and ($e[$k] != 0.0)) {
                        for ($j = $k + 1; $j < $nu; ++$j) {
                            $t = 0;
                            for ($i = $k + 1; $i < $this->n; ++$i) {
                                $t += $this->V[$i][$k] * $this->V[$i][$j];
                            }
                            $t = -$t / $this->V[$k + 1][$k];
                            for ($i = $k + 1; $i < $this->n; ++$i) {
                                $this->V[$i][$j] += $t * $this->V[$i][$k];
                            }
                        }
                    }
                    for ($i = 0; $i < $this->n; ++$i) {
                        $this->V[$i][$k] = 0.0;
                    }
                    $this->V[$k][$k] = 1.0;
                }
            }
            // Main iteration loop for the singular values.
            $pp   = $p - 1;
            $iter = 0;
            $eps  = pow(2.0, -52.0);
            while ($p > 0) {
                // Here is where a test for too many iterations would go.
                // This section of the program inspects for negligible
                // elements in the s and e arrays.  On completion the
                // variables kase and k are set as follows:
                // kase = 1  if s(p) and e[k-1] are negligible and k<p
                // kase = 2  if s(k) is negligible and k<p
                // kase = 3  if e[k-1] is negligible, k<p, and
                //           s(k), ..., s(p) are not negligible (qr step).
                // kase = 4  if e(p-1) is negligible (convergence).
                for ($k = $p - 2; $k >= -1; --$k) {
                    if ($k == -1) {
                        break;
                    }
                    if (abs($e[$k]) <= $eps * (abs($this->s[$k]) + abs($this->s[$k + 1]))) {
                        $e[$k] = 0.0;
                        break;
                    }
                }
                if ($k == $p - 2) {
                    $kase = 4;
                } else {
                    for ($ks = $p - 1; $ks >= $k; --$ks) {
                        if ($ks == $k) {
                            break;
                        }
                        $t = ($ks != $p ? abs($e[$ks]) : 0.) + ($ks != $k + 1 ? abs($e[$ks - 1]) : 0.);
                        if (abs($this->s[$ks]) <= $eps * $t) {
                            $this->s[$ks] = 0.0;
                            break;
                        }
                    }
                    if ($ks == $k) {
                        $kase = 3;
                    } elseif ($ks == $p - 1) {
                        $kase = 1;
                    } else {
                        $kase = 2;
                        $k    = $ks;
                    }
                }
                ++$k;
                // Perform the task indicated by kase.
                switch ($kase) {
                    // Deflate negligible s(p).
                    case 1:
                        $f         = $e[$p - 2];
                        $e[$p - 2] = 0.0;
                        for ($j = $p - 2; $j >= $k; --$j) {
                            $t           = hypo($this->s[$j], $f);
                            $cs          = $this->s[$j] / $t;
                            $sn          = $f / $t;
                            $this->s[$j] = $t;
                            if ($j != $k) {
                                $f         = -$sn * $e[$j - 1];
                                $e[$j - 1] = $cs * $e[$j - 1];
                            }
                            if ($wantv) {
                                for ($i = 0; $i < $this->n; ++$i) {
                                    $t                   = $cs * $this->V[$i][$j] + $sn * $this->V[$i][$p - 1];
                                    $this->V[$i][$p - 1] = -$sn * $this->V[$i][$j] + $cs * $this->V[$i][$p - 1];
                                    $this->V[$i][$j]     = $t;
                                }
                            }
                        }
                        break;
                    // Split at negligible s(k).
                    case 2:
                        $f         = $e[$k - 1];
                        $e[$k - 1] = 0.0;
                        for ($j = $k; $j < $p; ++$j) {
                            $t           = hypo($this->s[$j], $f);
                            $cs          = $this->s[$j] / $t;
                            $sn          = $f / $t;
                            $this->s[$j] = $t;
                            $f           = -$sn * $e[$j];
                            $e[$j]       = $cs * $e[$j];
                            if ($wantu) {
                                for ($i = 0; $i < $this->m; ++$i) {
                                    $t                   = $cs * $this->U[$i][$j] + $sn * $this->U[$i][$k - 1];
                                    $this->U[$i][$k - 1] = -$sn * $this->U[$i][$j] + $cs * $this->U[$i][$k - 1];
                                    $this->U[$i][$j]     = $t;
                                }
                            }
                        }
                        break;
                    // Perform one qr step.
                    case 3:
                        // Calculate the shift.
                        $scale = max(max(max(max(abs($this->s[$p - 1]), abs($this->s[$p - 2])), abs($e[$p - 2])), abs($this->s[$k])), abs($e[$k]));
                        $sp    = $this->s[$p - 1] / $scale;
                        $spm1  = $this->s[$p - 2] / $scale;
                        $epm1  = $e[$p - 2] / $scale;
                        $sk    = $this->s[$k] / $scale;
                        $ek    = $e[$k] / $scale;
                        $b     = (($spm1 + $sp) * ($spm1 - $sp) + $epm1 * $epm1) / 2.0;
                        $c     = ($sp * $epm1) * ($sp * $epm1);
                        $shift = 0.0;
                        if (($b != 0.0) || ($c != 0.0)) {
                            $shift = sqrt($b * $b + $c);
                            if ($b < 0.0) {
                                $shift = -$shift;
                            }
                            $shift = $c / ($b + $shift);
                        }
                        $f = ($sk + $sp) * ($sk - $sp) + $shift;
                        $g = $sk * $ek;
                        // Chase zeros.
                        for ($j = $k; $j < $p - 1; ++$j) {
                            $t  = hypo($f, $g);
                            $cs = $f / $t;
                            $sn = $g / $t;
                            if ($j != $k) {
                                $e[$j - 1] = $t;
                            }
                            $f               = $cs * $this->s[$j] + $sn * $e[$j];
                            $e[$j]           = $cs * $e[$j] - $sn * $this->s[$j];
                            $g               = $sn * $this->s[$j + 1];
                            $this->s[$j + 1] = $cs * $this->s[$j + 1];
                            if ($wantv) {
                                for ($i = 0; $i < $this->n; ++$i) {
                                    $t                   = $cs * $this->V[$i][$j] + $sn * $this->V[$i][$j + 1];
                                    $this->V[$i][$j + 1] = -$sn * $this->V[$i][$j] + $cs * $this->V[$i][$j + 1];
                                    $this->V[$i][$j]     = $t;
                                }
                            }
                            $t               = hypo($f, $g);
                            $cs              = $f / $t;
                            $sn              = $g / $t;
                            $this->s[$j]     = $t;
                            $f               = $cs * $e[$j] + $sn * $this->s[$j + 1];
                            $this->s[$j + 1] = -$sn * $e[$j] + $cs * $this->s[$j + 1];
                            $g               = $sn * $e[$j + 1];
                            $e[$j + 1]       = $cs * $e[$j + 1];
                            if ($wantu && ($j < $this->m - 1)) {
                                for ($i = 0; $i < $this->m; ++$i) {
                                    $t                   = $cs * $this->U[$i][$j] + $sn * $this->U[$i][$j + 1];
                                    $this->U[$i][$j + 1] = -$sn * $this->U[$i][$j] + $cs * $this->U[$i][$j + 1];
                                    $this->U[$i][$j]     = $t;
                                }
                            }
                        }
                        $e[$p - 2] = $f;
                        $iter      = $iter + 1;
                        break;
                    // Convergence.
                    case 4:
                        // Make the singular values positive.
                        if ($this->s[$k] <= 0.0) {
                            $this->s[$k] = ($this->s[$k] < 0.0 ? -$this->s[$k] : 0.0);
                            if ($wantv) {
                                for ($i = 0; $i <= $pp; ++$i) {
                                    $this->V[$i][$k] = -$this->V[$i][$k];
                                }
                            }
                        }
                        // Order the singular values.
                        while ($k < $pp) {
                            if ($this->s[$k] >= $this->s[$k + 1]) {
                                break;
                            }
                            $t               = $this->s[$k];
                            $this->s[$k]     = $this->s[$k + 1];
                            $this->s[$k + 1] = $t;
                            if ($wantv and ($k < $this->n - 1)) {
                                for ($i = 0; $i < $this->n; ++$i) {
                                    $t                   = $this->V[$i][$k + 1];
                                    $this->V[$i][$k + 1] = $this->V[$i][$k];
                                    $this->V[$i][$k]     = $t;
                                }
                            }
                            if ($wantu and ($k < $this->m - 1)) {
                                for ($i = 0; $i < $this->m; ++$i) {
                                    $t                   = $this->U[$i][$k + 1];
                                    $this->U[$i][$k + 1] = $this->U[$i][$k];
                                    $this->U[$i][$k]     = $t;
                                }
                            }
                            ++$k;
                        }
                        $iter = 0;
                        --$p;
                        break;
                } // end switch
            } // end while
        } // end constructor

        /**
         *    Return the left singular vectors
         * @access public
         * @return U
         */
        public function getU() {
            return new Matrix($this->U, $this->m, min($this->m + 1, $this->n));
        }

        /**
         *    Return the right singular vectors
         * @access public
         * @return V
         */
        public function getV() {
            return new Matrix($this->V);
        }

        /**
         *    Return the one-dimensional array of singular values
         * @access public
         * @return diagonal of S.
         */
        public function getSingularValues() {
            return $this->s;
        }

        /**
         *    Return the diagonal matrix of singular values
         * @access public
         * @return S
         */
        public function getS() {
            for ($i = 0; $i < $this->n; ++$i) {
                for ($j = 0; $j < $this->n; ++$j) {
                    $S[$i][$j] = 0.0;
                }
                $S[$i][$i] = $this->s[$i];
            }
            return new Matrix($S);
        }

        /**
         *    Two norm
         * @access public
         * @return max(S)
         */
        public function norm2() {
            return $this->s[0];
        }

        /**
         *    Two norm condition number
         * @access public
         * @return max(S)/min(S)
         */
        public function cond() {
            return $this->s[0] / $this->s[min($this->m, $this->n) - 1];
        }

        /**
         *    Effective numerical matrix rank
         * @access public
         * @return Number of nonnegligible singular values.
         */
        public function rank() {
            $eps = pow(2.0, -52.0);
            $tol = max($this->m, $this->n) * $this->s[0] * $eps;
            $r   = 0;
            for ($i = 0; $i < count($this->s); ++$i) {
                if ($this->s[$i] > $tol) {
                    ++$r;
                }
            }
            return $r;
        }
    }
